Cooper-Pair Box (Charge Qubit)

Figure

Description

A nanometre-scale superconducting electrode (the “island”) connected to a reservoir via a Josephson junction constitutes an artificial two-level electronic system: a single-Cooper-pair box. The two levels consist of charge states differing by one Cooper pair ($2e$) that are coupled by tunneling through the junction. Although the two-level system is macroscopic, containing a large number of electrons, the two charge states can be coherently superposed.

The Cooper-pair box operates in the charge regime where $E_C>E_J$ — the electrostatic energy cost of adding a Cooper pair to the island exceeds the Josephson coupling energy. At the charge degeneracy point $n_g=C_g{V}_g/2e=1/2$, the two lowest charge states $|n⟩$ and $|n+1⟩$ are degenerate and split by the Josephson energy $E_J$, forming a protected sweet spot with first-order insensitivity to charge noise. Away from this point, the qubit frequency is strongly sensitive to gate charge fluctuations, which historically limited coherence times.

The Cooper-pair box was the first superconducting qubit to demonstrate coherent quantum oscillations (Nakamura, Pashkin, and Tsai, 1999), establishing superconducting circuits as a viable platform for quantum computing. Its extreme charge noise sensitivity motivated the development of the transmon ($E_J/E_C\gg 1$), which trades anharmonicity for exponential suppression of charge dispersion.

Hamiltonian

$H=4E_C(\hat{n}-n_g{)}^2-E_J\cos \hat{\phi }$

where $E_C=e^2/2C_{\Sigma }$ is the charging energy, $C_{\Sigma }=C_J+C_g$ is the total island capacitance, $n_g=C_g{V}_g/2e$ is the gate-induced charge in units of Cooper pairs, $E_J$ is the Josephson energy, and $\hat{n}$, $\hat{\phi }$ are conjugate charge number and phase operators.

In the charge regime ($E_C\gg E_J$), the two lowest-energy states near the degeneracy point $n_g=1/2$ are approximately:

$H_{\text{2-level}}\approx -2E_C(1-2n_g){\sigma }_z-\frac{E_J}{2}{\sigma }_x$

The qubit frequency ${\omega }_{01}=\sqrt{(4E_C(1-2n_g){)}^2+E_J^2}/\hslash$ is minimized at degeneracy (${\omega }_{01}=E_J/\hslash$) and is first-order insensitive to charge noise $\delta n_g$ at that point.

Motivation

The Cooper-pair box was the original “artificial atom” formed in a superconducting circuit, demonstrating that macroscopic electrical circuits could exhibit quantum coherent behavior. Its sensitivity to charge noise ($\delta n_g$) at generic operating points motivated the development of charge-insensitive variants: the transmon (large shunt capacitance, $E_J/E_C\gg 1$) and the fluxonium (superinductive shunt). The CPB remains foundational as the circuit from which the entire family of superconducting qubits descends.

Experimental Status

First coherent oscillations — Nakamura, Pashkin, and Tsai (1999):

• Demonstrated time-domain coherent quantum oscillations in a single-Cooper-pair box.
• Gate-voltage pulses drove Rabi-like oscillations between charge states.
• Coherence limited by charge noise to $T_2\sim 1\text{ns}$.

Quantronium — Vion et al. (2002):

• Operated at the charge degeneracy sweet spot, achieving $T_2\sim 0.5\mu \text{s}$.
• Introduced Ramsey interferometry and spin-echo techniques for superconducting qubits.
• Demonstrated single-qubit gate fidelity ~99%.

Two-qubit coupling — Yamamoto et al. (2003):

• Capacitive coupling between two CPBs demonstrated with ~95% gate fidelity.

Key Metrics

Metric	Value	Notes	Fidelity reference
$T_1$	~1–10 μs	Limited by charge noise	Nakamura et al. 1999
$T_2$	~0.5–5 μs	At sweet spot ($n_g=1/2$)	Vion et al. 2002
1Q gate fidelity	~99%	Voltage-pulse driven	Vion et al. 2002
2Q gate fidelity	~95%	Capacitive coupling	Yamamoto et al. 2003
Gate time (1Q)	~1–10 ns	Fast voltage pulses	—
Gate time (2Q)	~10–50 ns	Capacitive coupling	—
Readout fidelity	~90–95%	Probe junction or SET	Vion et al. 2002
Qubit footprint	~1 × 1 μm²	Very compact	—
Operating temperature	20–50 mK	Dilution refrigerator	—
Connectivity	Fixed capacitive	Nearest-neighbor	—

References

Original demonstration

• Y. Nakamura, Yu. A. Pashkin, and J. S. Tsai, “Coherent control of macroscopic quantum states in a single-Cooper-pair box,” Nature 398, 786 (1999)

Key experiments

• D. Vion et al., “Manipulating the Quantum State of an Electrical Circuit,” Science 296, 886 (2002)
• T. Yamamoto et al., “Demonstration of conditional gate operation using superconducting charge qubits,” Nature 425, 941 (2003)

Related theory

• V. Bouchiat et al., “Quantum Coherence with a Single Cooper Pair,” Phys. Scr. T76, 165 (1998)

Linked Papers

• nakamura-1999-cpb-coherent-oscillation

Related Entries

• transmon — descendant operating in $E_J\gg E_C$ for charge noise immunity
• fluxonium — inductive shunt alternative to capacitive shunting
• blochnium — quasicharge regime of the fluxonium circuit
• phase-qubit — operates in the complementary phase regime
• flux-qubit — encodes in persistent-current states rather than charge
• circuit-qed — CPB was the original qubit in the Blais et al. proposal